Providing the statistical QoS objectives in high-speed networks

and ISON

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Computer Networks and ISDN Systems 29 (1997) 1919-1931

Providing the statistical QoS objectives in high-speed networks Hoon Lee a,* , Yoshiaki Nemoto b a Switching

Technology b Graduate

Research Laboratories, SchooI of Information

Korea Telecom, 17, Umyon-Dong, Socho-Gu, Seotd 137-792, South Korea Sciences. Tohoku University, Aoba, Aoba-ku, Sendai 980, Japan

Abstract We present an approximate analytic model for providing the statistical QoS (Quality-of-Service) objectives in High-Speed Networks. First, we derive approximate formulae for the tail distribution of the queue length and the cell delay for the discrete time G/D/c queueing system with correlated input arrivals. Next, we present the relationship between the tail distribution of the queue length with the required service rate in the statistical aspects. Next, we present the tail probability of loss period ar a statistical QoS measure. We assume the statisticalmultiplexerof the ATM outputport with cell admission

regulationsbasedon the arrival processes, andinvestigatethe performances. Finally, we verify the accuracyof the theoryvia experiments of the numerical computations and simulations. Keywords:

0 1997 Elsevier Science B.V.

Statistical Quality-of-Service; Tail distribution; Delay loss; Delay loss period; ATM output multiplexer

1. Introduction

The typical Quality-of-Service (QoS) measures for the services in ATM networks are the cell loss and delay. For the applications having real time constraints, the cell delay such as the maximum tolerable delay to transmit a cell acrossthe network plays more important role. In other words, if a deadline is associatedwith each cell and if the it is supposedto reach its destination after its deadline is passed,it ma.y be considereduselessand is subject to be discarded a priori [1,2 I]. We denote this loss as the delay 10s.~. Typically, the QoS values are specified on an end-to-end basis.However, the exact solution to the end-to-end delay is hard to obtain. Thus, the appor* Corresponding author. E-mail: [email protected].

tionment of the end-to-end QoS values to the local QoS values in a node is usually adopted [21]. So, in this paper, let us confine our interests in an elementwise delay loss in a switch, and determine that an arriving cell is discarded if it finds the queue length to be the greater than the predefined value. As for the guaranteeof the delay QoS, there exist two important problems that should be addressed. One is the input regulation for the graciousdegradation of the delay loss. This has a significant effect on the performance of the overload-prone case, especially in the ATM multiplexer with high load which accepts highly bursty arrivals 1181.Because, for the real time services such as voice and video, if the system has a possibility to fall into heavily loaded state, some portion of the cells can be discarded without sacrifices in the QoS felt by a user. By doing that, the QoS will degrade smoothly (thus, gra-

0169-7552/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO169-7S52(97)00103-7

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ciously), otherwise heavy degradation of the QoS may occur by elongated overflow states. Another is the measure that how long the hazard state such as the delay loss period lasts [ 18,191. The measure of loss period length has another important meaning in the system performances if the interests lie in the time correlated real time cell streams. It is known that there are two major types for the QoS guarantee: the deterministic guarantee and the statistical guarantee [7]. Among them, the statistical QoS guarantee is considered to be more suitable in investigating the probabilistical behavior of the QoS and is recognized as more realistic method for solving the above mentioned problems. Therefore, in this paper, the statistical QoS measure is considered and it is defined as follows. The probability that the delay experienced by a cell is larger than a specified target value does not exceed some value required by a service. That is, the statistical QoS measure is formulated as follows [7]: Prob(delay 2 target value) I bound. The researches into the tail distribution of the queue can be found from several literature [5,8, 14,15,25]. On the other hand, the following literature deal with the statistical QoS measures for the ATM networks. Ferrari [7] and Kurose [16] defined the general concept of the statistical QoS guarantees. Roberts and Virtamo [23] proposed upper and lower bounds for the queue length distributions for the superposed periodic cell arrivals via non-exponential approximation functions. Hui [12] had applied the large deviation techniques to evaluate the probability of burst blocking in the broadband networks. Kesidis et al. [13] derived the approximate formulae of the queue length distribution for the Markov fluid source and the Markov modulated sources via heuristic method. Guerin et al. [9] made comprehensive discussions for the approximate analysis to the equivalent capacity for the networks, whose concept is similar to the statistical QoS measures. Chang [3] proposed a mathematical model for the deterministic upper bound for the tail probability of the queue length and delay of the G/D/c queue in discrete-time and discussed their stability condition for the cell arrival process called the envelope process. Yaron and Sidi [26] proposed a model for the upper bound of the queue

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length using exponentially bounded arrival process by imposing exponential decay in the distribution of the burst length. However, Yaron and Sidi’s model assumed the coefficient of the exponents to be constant with respect the decay rate, and there is no explicit formula for the decay rate. There are many works which deal with the cell acceptance control for the statistical multiplexer in view of the mean performances 16,171. However, if it is concerned with the statistical QoS measure, we could not find any work yet to our best knowledge. Thus, in this paper, we discuss the statistical QoS guarantee problems by modelling analytically the tail probabilities for the cell delay loss and its period, and investigate its performance by considering the cell acceptance control. First, based on the concept of the large deviation techniques, we derive an explicit closed-form formula for the upper bound of the tail distribution of the queue length and delay for the general dependent arrivals and deterministic service system. Next, we will apply those results to the problem of statistical QoS guarantee which takes into account the cell acceptance control for the output multiplexer of the ATM networks. The remainder of this paper is composed as follows. In Section 2, we present some preliminaries concerning approximate upper bounds for the tail distribution of the queue length for the general G/D/c queue with correlated arrival, from which we derive the upper bound for the tail distribution of the cell delay. In Section 3, we give the formulae for the statistical QoS measures such as the mean service rate which guarantees the statistical cell delay requirements, and the tail probability of the delay loss period by applying the results of Section 2. In Section 4, we present the numerical results via computations and simulations and, give some discussion. Finally, in Section 5, we summarize the paper.

2. Preliminaries

2.1. Queue model Let us consider a discrete-time G/D/c which is a typical queue model in ATM

queue context.

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Suppose that the time is divided into slots of unit length during which a burst of cells arrives, and at most c cells depart the queue just before the end of time slot. The cell egress follows the FIFO principle. For the cells anrived simultaneously in a slot, a cell egresses in random order. Let us assume that the arrival process igenerates a batch which is dependent on the state of the source, and it is correlated with respect to the time slot. The arrival process is governed by a modulating process G = {Gk. k = 1, 2, . . .}, which we call a phase process. G, is assumed to be a Markov chain on the state space A={l,2, . ..) IM} with the transition matrix P has an element pij = Pr{Gk+ , = jlG, = il. Let A,(G) = k:= 1,2, . . . } be a sequenceof random i A’,, i&H’, variables, which describesthe number of cells that arrive during time slot k when the source’s phaseis G, = i for A;. Now let us assumethat the input regulation is carried out as follows: The input controller regulates the ingress of the cells generated by a source based on the phaseof the source, and it is governed by the following function: Yl = f( A;), that is, among the A’, cells arrived in a slot k when the modulating processis in phase i, only Yk cells can be admitted into the queue, while the remaining (A; - Y,‘> cells are discarded. Let the random variable Xi be the queue length at the beginning of time slot k given that the phaseis i at that time slot. The system state evolves at the boundariesof the time slot. Note that, in the sequel, the superscript indicates the phase of the arrival process, which is omitted without loss of generality when it is possible, while the subscript indicates the time slot index and its related representation. We assumethat X, = 0, that is, the system is empty at the initiation of a slot. Then, we Ican represent the number of cells remained in the queue at the beginning of time slot k,

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+ Yk-, -c,O].

(‘1 Fig. 1 illustrates the queue evolution between the time slots. 2.2. Upper

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yk-l yr*7 yk i {{ .,,4 + ’ (k-l)-th

slot

I k-th slot

-) Time

’

C b 4 Fig.

I. Queue

evolution

between

the time

slots.

first have to derive the formula for the upper bound of the queue occupancy distribution. As an approximate method for the upper bound of the queue occupancy, the large deviation techniquesare considered to be a useful method [2,41. Referring to Eq. (I), let us denote the variation of the queue length in a time slot k to be 2, = Yk- c. We also denote a function for the cumulative variations between time slot [k,l] as Z,,, = C:,,Z,, and let L,,,(B) = (l/m)logE[exp(OZ, ,I], where m = I -k+ l,m>O. We denote a limit by L(O)= lim m-r~ L,(8), t/8 E R, where R is a positive real set. L(B) is assumedto be strictly convex ’ 1241and differentiable for 0 < fI < 3~. The convexity of the function L( 0) can be easily proved by using Halder’s inequality (101. Then, for E> 0, we can assumethat there exists mE such that for all m > m, E[exd%.d]

~ex~[V0)

+

+I.

(2)

From Eq. (1) we obtain X, = Max,,,-,+,Z,~,, and if we use the inequality Max(x, y) I x + y, for x,y 2 0, then we obtain E[exd WJ]

2 i E[ev( m=l

f%..J]

m-1

+ m.m, C ed(L(@)+E)ml. (3)

X,7 ‘v

X,=Max[X,-,

1919-I

If I,( 0) < - E holds, then we have E[exp(

ex,)]

s C( e> < r,

bound for tail distribution

I In order to obtain the upper bound of the probability that the cell delay exceeds a certain value, we

g(.) q(x)+(l--

UE[O,ll.

is said to be convex if and only if ~(a.x +(I - a)~)< u)g(y) for alI x.y E I, I is a set of integers, and all

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where

C( 0) = ? E[expWdl.

(4)

m=l

In order to estimate the upper bound of C(e), let and define n(e,m) = sup,(l/0)log Q,,(e). If there exists A(B) which satisfies A(B) < c, and an arbitrary small real valued s(e) such that n(e,m> = (A(B) - c)m + S(e), then we have

f&,,(e) = E[exp(eZ,,,>l,

lim - c = m-x

w

Q(e,m)

-s(e)

W) = sup 8 e

m

’

and

s(e)=

SUP su&logn,(ej mz1[ e

-L(e)m)

1

. (5)

In order to obtain a closed formula for c(e), we take the summation over = in formula (41, which results in a rough upper bound. That is,

c(e)5

exp[e((A(e)-++6(e))]

i m=l

= exp[e(A(e) --c+ 1 -exp[e(h(e) Finally, from the Markov Pr( X, 2 x} = Pr{exp( I

%@))I -c)]

’

(6)

sc(e)exp(-Ox),

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length x, this can be induced from the result of formula (7), since, in that case, the server serves c cells every time slot almost surely. Thus, for the case of sufficiently large x (x B- c> and t, the delay limit imposed upon the cell, we can consider that the tail probability of the delay experienced by a cell has a mean with the same distribution as that of queue length. We will prove this fact. From Eq. (I), we can deduce that, as we had assumed the randomness among the cells that arrived simultaneously in a slot, the minimum delay that is experienced by a cell that arrived during the time slot k (we call it just a delay henceforth) is upper-bounded by [ XJcl, where [ yl indicates the smallest integer greater than or equal to y. That is, the tail probability that the cell delay exceeds t time slots is equivalent to the probability that the queue length exceeds x, where t = [x/cl. So, we obtain the following upper bound: Pr +2t i

IPr(X,2c(d-

1)).

1

Thus, if we denote W, to be the minimum delay of a cell which arrives during time slot k, then we have the tail probability for the cell delay given by

sc(e)exp(--ce(t--

1)).

(9)

inequality [ 111, we have 2.3. Decay rate

’

from which we obtain

pr{x,2~}

Systems

Pr(W,lt]

ex,) 2 exp( 8x))

E[exp( ex,>l exp( Ox)

and ISDN

(7)

where 8 is the largest solution of the inequality L(d) < 0. We call 0 the decay rate and C(0) the prefactor. Inequality (7) holds for the stationary distribution as well if the sequence {Z,,} is stationary and ergodic. We prove this in Appendix A. Now let us consider the tail probability of the delay. We define the delay as the number of time slots an arriving cell has to wait till it receives the service. For the FIFO server, the delay depends on the number of the cells waiting for the service in the queue. For the case of a sufficiently large queue

In order to obtain a formula for the decay rate of the approximate formula (7), let us consider the detailed behavior of the admitted arrivals. Without loss of generality, we call the admitted arrivals just arrivals. We define the moment generating function (mgf) of the arrival process by 9 ‘( 0) = E[exp(B Y,i)]. Let us denote a diagonal matrix U(0) by q(e) = diag(ll”(e), q*(e), . . . , lu”(e)), where diagc.1 indicates the matrix composed of diagonal components given by (.> while the other elements are equal to zero. Let a matrix F = ly(0)P. We denote g(F) to be the spectrum of F. The spectrum of F is defined to be the set of all r E R that are eigenvalues of matrix F. Let sr(F) be the spectral radius of the matrix F defined by sr(F) = max{lrl: T E (T(F)}. In order to derive the relationship equivalent to the stability condition, let us define a limit function

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by M(B) = lim,,, (1 /m) log E[exp( 8 Yk,,)], where Y,,, = C{=,Y,. Then, we have two results with respect to M(8), which is given as follows: 1. M(O) is upper-bounded such that M(8) log sr(F).

I

Proof. For the arrival processwith generator G. we have the following formula. E[exr@ Y,.,J] (10) where Y,,,+ is the number of arrivals at the end of time slot m given that the Markov chain was in state i at the initial time slot, and pi, indicates the transition probability of the Markov chain from state i to state j. Let @,,(B > = (E[exp{O Y,,,,,,)l, . . ,E[expl~ Y,.+, )]) and let @,,,(0)T be its transpose. If we write Eq. (10) in matrix form, it is given by @,&3)T=F@:,_,(B)T.

= ~@k(fvTT

which in turn becomes E[exp{BY,,,]]

= ~(F)“-‘~(0)IT.

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that is, P” > 0 for n 2 1. As a result, the matrix F is primitive becauseV’(0) is positive. In order to derive the final result, we introduce the PerronFrobenius theorem for the eigenvalue and its eigenvector, which is summarizedin Appendix C. Thus, based on this theorem and from Eq. (12), the assertion2 is proved. 0 Finally, we can obtain the decay rate from the stability condition of M(B) < 4. That is, we have 0 which is a largest solution of the inequality fl> ilugsr(F).

(13)

3. Statistical QoS measures By using the result of Section 2, we consider, the statistical QoS (SQoS) measurefor an ATM switch. 3.1. SQoS and required

service

rute

(1’)

An initial condition is given by @,(OjT = !?!(8)IT, where I is a raw element matrix. Now let 7~~be the initial probability that the Markov chain is in state i, and its vector is given by x = (7~,, . . ,nTTM).Then, from the property of the conditional probability, we have E[exp{eY,.,:t]

und ISDN Systems 29 (1997)

(12)

From the property of the convergence of the matrix norms with respect to the spectral radius [lo], the matrix (FY” IS upper-bounded by p(BXsr(F) + elm, where p(0) and E are arbitrary positive constants. This is proved in Appendix B. Then, from Eq. (12). the assertion 1 is proved. Cl 2. If the Markov chain is irreducible and aperiodic, then the equality holds, that is, M(8) = logsr(F). Proof. Since we had assumedthe ergodicity of the Markov chain, the transition matrix P is primitive,

Let us consider the relationship between the SQOS required by a service and the corresponding system capacity. Given the arrival processesand the SQoS constraints defined by the upper bound on the probability distribution of cell delay, we estimate the mean required service rate. We denote the effective service rate as the mean required capacity of the server in equilibrium for the output multiplexer of a switch to fulfill the required SQoS. The notion of effective service rate is similar to that of the effective bandwidth which is defined in [9]. The effective bandwidth is the minimum mean service rate required by a system to guarantee a QoS, e.g., the buffer overflow probability, defined by an aggregation of sources. Then, for the deterministic server we had assumed, the SQoS for the delay loss t is mapped to the probability that the queue length X, where X = et, of an infinite capacity queue exceeds a specified value K should be below a certain value K. Alternatively, Pr(XtK}

SK,

(14)

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where K is an integer and K is a real number. Thus, if we relate the inequality (14) with the inequality (71, we obtain Pr{X>K}
(15)

Then, we obtain the following

equation Time Slot (16) ’ Delay Loss Perio

which can be solved method.

by numerical

root finding

Fig. 2. Sample path of delay loss period.

3.2. A distribution for delay loss period Based on the definition of SQoS measure, let us assume that an arriving cell is lost when it finds that the queue length is greater than or equal to a threshold value K. This implies that the arriving cell can not be guaranteed its delay requirement if the queue length is not smaller than K. Thus, it is lost a priori. We call the period during which the queue length is greater than or equal to a threshold value K as a delay loss period. In order to obtain a distribution for a delay loss period (DLP), let us investigate the evolution of the queue length with respect to time slot, which is illustrated in Fig. 2. Let us divide the DLP into two time periods; one is the period composed of time slots that have passed already at the observation point, the time slot k, and the other is the period composed of the time slots that may last from that instant. We call each of them the trace and the residue, and is denoted by h, and r,, respectively. Let us first consider the trace. The trace of DLP observed at time slot k is described as follows: h,=Min[j:X,>KIX,-,-=K]. We approximate holds: Pr{ h, =j}

that the following

I Pr{Xkmj = K)Pr(Yk-j.k

(17) upper bound ljc}-

(18)

Then, the probability that the length of the trace is not smaller than 1 is given by Pr{h,rl)

= iPr(h,=j}. j=i

(19)

Let us rewrite the formula (18) into matrix form by taking into account the phase of the source at the observation point. In order to do that, let us define some variables. Let h,(j) = (hi(j) hi(j) . . . hr( j)) be the matrix composed of the probability h;(j) = Pr{ h, = jlsource phase = i). Let us write the conditional probability of queue occupancy which is defined explicitly in Appendix D as follows: Xk-j(

K) = ( Xi-,(

K), . . . TX,“j( K))T.

(20)

Let us denote E= diag(w, w, . . . , w), where w = 2 jd. Then from (18) and (20), we obtain Nyk-j,k

x,(j)

I ,ykmj( K)TE.

w

where x~- j( I( jT is the transpose of the matrix x/I _ jfK). Note that, in Appendix D, we assumed that as the queue length is not smaller than the threshold value K, the arriving cells are assumed to be rejected, which gives us an equivalent finite state queueing system with state variable ranging {0,*, . . . , K}. Note that this assumption is restricted only to the derivation of Eq. (34), otherwise we relax this assumption. Finally, the probability that the trace is not smaller than 1 is given by i,( 1) = i&(j) j-l

I ~ X~-j( K>‘a. i-1

(22)

Now let us consider the residue of DLP. The residue of DLP seen by an arbitrary observation point k is given as follows: r,=Min[i:

X,,,


(23)

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Since the residue is conditioned on the trace, we have Pr{r,li)=

i,Pr{(h,=j)fl(r,zi)}.

(24)

j=O

and

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where the explicit formula for the first term in the right hand side is obtained by mathematical induction between the probability mass function and its tail distribution, to be Pr{X = K) = C(0Xl ewe)ewBK.

If we use an independenceapproximation between the two events, we obtain 4. Performance evaluation Pr{r,li}

I f,X,-JKjTZZ*,

(25)

j=O

where E*=diag(w*,o*, . . ..w*) and o* = Pr{Y,,,+i 2 ic}. Let us rewrite the formula (25) into Tk(i)l

~x~-~(K)~SS*,

(26)

j=O

where F,(i) = (r:(i), r:(i), . . . , rkM(i)). The DLP, which is observed at time slot k and will iast 1 time slots in total is. given by DLP, = h, + rk = f, and its tail probability is given by Pr{ DLP, 2 I} := Pr{ h, + r, 2 I}

We present the results of the performance concerning the SQoS measuresvia the numerical computation and the simulation experiments. We assume various source cases; the binomially distributed source as an example of time independent bursty source, and the On/Off source as a time correlated two-phase source, and the three-phaseMarkov modulate source as that of more complex time correlated source. First, we give the upper boundsfor the tail distribution of the cell delay loss probability. Next, we present the estimation of the mean service rate required to guarantee the delay loss SQoS. Finally, we illustrate the results for the loss period distribution. 4.1. Source model

(27)

Let us denote m,(I) as m,(I) = (DLPL(I), DLPl(/>, _. _, DLP:(/)), where DLP$I) = Pr{DLP, 2 [(G, = i). Then, we have an approximate formula for the upper bound of the probability that the delay loss period is not smaller than 1 time slots given by

First, let us assumea three-phasesource asshown in Fig. 3. If we describe its phasetransition in matrix form, it is given by

P= [

where 8= diag(3, ij, . . . , B) and 3= Pr{y,-j 2 (I - j)c}, where y, indicates the mean cumulative arrivals over 1 time slots, which meansthe mean of the total number of cells arrived during the I time slots. Now let us consider the approximation of the upper bound for the stationary probability of the loss period. We do this heuristically by observing the formulas (7) and (181, from which we obtain the stationary upper bound for the lossperiod as follows: Pr(DLP>t)

rPr{X=K}Pr(y,2Ic},

(29)

1-a-p 6

ff 1-6-y

rl

5

P Y l-77-5

.

(30)

I

We assumethat, in phase i, i = 1, 2, 3, the source generatescells with arbitrary rate function r,.

1-c-p c-l co#o% 1

1-E-r

2 -3 -5

Fig. 3. State transition

Y

diagram

11

03

for three-phase

I-0-l

source.

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Second, let us consider an On/Off source. The state transition matrix of On/Off process would be given as follows: 1 - 92

P= [

91

q2 l-9,.

I

(3’)

The modulating chain has two states, state 1 and 2, and cells arrive with rate r, and r2 for each state, respectively. In each time slot the phase will changes from state 2 (or 1) to state 1 (or 2) with probability 9t (or q2). Otherwise, the phase remains in state 2 (or 1) with probability 1 - 4, (or I - q2). Finally, let us assume an arrival process which has just a single phase; and the cell arrivals are rene’walwith probability p [20]. Returning to the three-phase source, let us give the regulation function for k > 1 as foliows:

Yk=

rl v2 5r3

if G,= 1, if G, = 2, if G, = 3.

(32)

The diagonal matrix of the moment generating function for the arrival process is ‘I’ = diag(e”l, ew> , ee5’l). 4.2. Tail probability of cell delay loss Let us first consider the renewal arrival: An aggregation of N connections constiiutes a batch of cell arrivals, and we assumethat the number of cell arrivals in a slot is distributed binomially: That is, the parametersfor the cell arrivals are given by a set (N, p) and N, the number of associatedconnections, is assumedto be N = 20. The probability p has three values 0.045, 0.040, and 0.035 which correspond to the offered load of 0.9, 0.8 and 0.7, respectively, when the service rate c = 1. We denote the three parametersetsas P 1 = (20,0.045), P2 = (20,0.040), and P3 = (20,0.035), respectively. Using these parameters and from the formula (13), we obtained the decay rate, ~9,given by 0.233, 0.521, and 0.888 for each parameter set PI, P2, and P3, respectively. In order to obtain the decay rate with less computational complexity, we approximated the binomial distribution into the Gaussian one [20] and used the Legendre transform, from which we calculated the decay rate.

‘e-2010

20

30

40

50

60

70

60

90

H)

t, Time Slot Fig. 4. Delay loss for binomial

source.

Fig. 4 illustrates the upper bound for the tail distribution of the cell delay loss probability. The real lines indicate the results obtained by the numerical computations, and the marked points indicate those obtained by simulations. This notation is applied to all the casesin the sequel unless explicitly noted. From the results, we can find that the approximate analytical results are in good agreement with the simulational results. As for the On/Off source, we can obtain an explicit formula for the eigenvalue from the quadratic equation, which is well known (for example, refer to DOI). We assumed two cases of parameter set E = (9,,q2, r,, r2): that is, El =(0.5,0.4,0,2) and E2 = (0.65,0.35,0,2). The service rate is assumedto be one cell in a slot. For these parameter sets El and E2, the mean cell arrival rates ,are obtained to be 0.8894 and,0.699, and the decay rates are obtained to be 0.1823 and 0.6190, respectively. Fig. 5 illustrates the results for the tail distribution of the cell delay loss of On/Off source.

t, Time Slot Fig. 5. delay

16~s for On/Off

source.

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I

20 f

10

I 20

30

40

t, Time Slot

50

Threshold

Fig. 6. Delay loss for three-phase

source.

Fig. 8. Mean required

Finally, as to the Markov modulated three phase source with a parameter set T= (a, /?, 6, y, q, 0, we assume two cases: a source with relatively even phase transition (Tl) and a source with elevating variation between the states (T2): Tl = (O&0.3, 0.3,0.4,0.5,0.3) and T2 = (0.6,0.1,0.2,0.7,0.6,0.2). The arrival parameter has a set ( r, , r2, r3> = (1,2,3). The service rate is assumed to be c = 2. The stationary distribution for the two sets, Tl and T2, were obtained to be (0.328,0.366,0.306) and (0.365,0.315,0.321), respectively. The mean arrival rate for the T 1 and T2 are given- by (1.980, 1.956). The decay rate for each parameter set Tl and T2 are obtained to be 0.1091 and 0.1978, respectively. Note that, for the case with phase size greater than two, there is no explicit formula for the eigenvalue of matrix F defined in Section 2.3. It is obtained only by numerical root finding method. Fig. 6 illustrates the results for the upper bound of the cell delay loss rate for the parameter set T2. As we

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60

70

80

90

100

K

service rate for On/Off

source

can see from the result as well as in the previous two experiments, the approximate upper bound has a good agreement compared with the time consuming simulation result. This indicates that the proposed formula can estimate well the cell delay loss probability with simple and less expensive computational costs compared with the time consuming standard simulation. 4.3. Required service rate Now let us compute the mean service rate required by a server in an output multiplexer to satisfy the statistical QoS requirements. We assumed the same three arrivals as in Section 4.2. Here, the parameters defined in the previous subsection will be assumed for each source unless it is newly defined. First, let us consider the binomially distributed source. The target SQoS requirements for the cell delay loss are assumed to range from 10e6 to 10d9.

2r Target a, .o

1.6

g

1.4

.$

1.2

S-QoS=

16’ for all K

2

H E

' 0.8 1. '%

1.5' 20

30

40

50

Threshold Fig. 7. Mean

required

60

70

80

90

100

10

20

30

K

set-vice rate for binomial

40

50

Threshold source

Fig. 9. Mean required

60

70

60

90

I loo

K

service rate for three-phase

source

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zi I-

le-07

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I-

2

3 I, Loss Period

4

5

Fig. 7 illustrates the mean service rate for system with the binomially distributed source. The threshold is assumedto vary from 10 to 100 and the upper group, the group P 1, of the curves indicates the case when the average cell arrival rate with p = 0.045, while the lower one, the group P3, indicates that of p = 0.035. We found that the required service rate for more severe QoS application is greater than that of the less severe one. For example, for the queue size of 10, the required service rate is 2 times greater than its mean arrival rate. This becomessmaller as the queue length becomeslarger. Second, let us consider the On/Off source. The target SQoS requirementsare assumedto range from 10m6to 10e9. Fig. 8 illustrates the result. The upper group is the caseof El, whereasthe lower group is the case of E2. For the same queue size and the average cell arrival rate, we can find very similar trend in the mean service rate. Finally, let us consider the three phasesource.We assumethe following parameter set for the access 0.1

1

K=lO

a” I s

KG0

2

lem06

2-

3

4

5

I, Loss Period

Fig. 10. Probability of loss period for binomial source

5

and ISDN Sysrems 29 (I 997) I91 9-1931

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3 4 I. Loss Period

5

Fig. 11. Probability of loss period for On/Off source.

Fig. 12. Probability of loss period for three-phase source.

regulation: (cp, 6 I= (O&0.7). While, the target SQoS requirement is assumedto be IO-‘. Fig. 9 illustrates the result for the required service rate. Note that the effect of the input regulation becomes noticeable as the threshold becomessmaller. 4.4. Probability of delay lossperiod We investigate the time period the queue is in lossy state. The parametersused are the same as in the previous section unlessit is defined explicitly. Fig. 10 illustrates the tail probability distribution for the loss period when the input is a binomially distributed source. The lines indicate the analytical results, whereas the points indicates those obtained by simulational experiments. In Fig. 10, we assumed that the mean offered load of 0.9, which corresponds to the superposedindependent sourceswith the parameter (20,0.045). The threshold is assumedto be K = 10,20, and 30. The arriving cells are discarded if they find that the queue length is not smaller than the value K when he arrived to a queue. The abscissaindicates the consecutive time slot of loss period varying from 2 to 5. We found that the greater the threshold is, the closer the approximation toward the simulation results are, which we had expected. Fig. 11 illustrates the results for the G/D/I queue with the On/Off source as an input. From the results, we knew that the two results (obtained by computation and simulation) have a good agreement for the case of K = 20 and K = 30, whereas it overestimateswhen the queuelength is K = 10 which is consideredto be small.

H. Lee. Y. Nemoto

/ Computer

Networks

Fig. 12 illustrates the results for the G/D/2 queue with the three-phase source as an input. We investigated the result for the thresholds K = IO, 30.

and ISDN Systems 29 (1997)

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surely), and such that lim, -f li Pr(U, = U, n 2 m) = 1, for any initial value U,. Then, we finally have Pr{ X, = U, n 2 m} = 1, for any initial value lim,,, Xl.

5. Conclusion Appendix B. Upper bound for a primitive matrix We proposed an approximate analytical model for measuring the statistical Quality-of-Service (SQoS) in the high speed networks, in particular, for an ATM switch node. We first derived upper bounds for the tail distributions of the queue occupancy probability and cell delay loss problability. As an application of the analytical model, we considered the effective service rate of the server which satisfies the required statistical QoS. Next, we derived an approximate formula for the consecutive time period during which the queue length exceeds the threshold. From the numerical experiments, we obtained the following results: The performance of the statistical QoS obtained by the approximate model agrees very closely with the results from the simulation. This illustrates that the proposed approximate model could be used as a simple method to estimate the statistical QoS measure for the ATM switch.

Acknowledgements The authors would like to thank the anonymous reviewers for their comments. Their remarks improved the accuracy and presentation of this paper.

Appendix mula (7)

A. The stationarity

discussion

of for-

The stationarity condition for the sequence {Z,,) is that E[Z,] < 0 as n -+ x~ If we assume that there exists a stationary ergodic sequence (u,} such that Pr{u, =Z,, V n 2 m} = 1. Let {U,} be a lim,,, sequence sati,sfying the recursion U,, , = Max[ Urn + urn,0], for n 2 1, where U, 2 0 is a finite random variable. Then, since the sequence{u,} is stationary and ergodic, there also exists, E[u] < 0, and there exists a stationary sequence{U,} with generic element given by X, such that X < m a.s. (almost

For a given n X n matrix F, let us assumea real constant E> 0. Then, there is a constant p = p(F, E) such that

I(F Sp(sr(F)+ l )” for all k = 1,2,. . . , and ij = 1,2,. . ,n. Proof. Since the matrix F = (sr(F + E)-’ F has a spectral radius strictly less than 1, it is convergent as., and hence F” + 0 as k + x. In particular, the elementsof the sequenceF’ are bounded, so there exists a finite p > 0 such that IF,;15 P for all k= 1,2,...,

and i,j=

1,2,..., n. 0

Appendix C. Perron-Frobenius

theorem

Let us summarize the Perron-Frobenius theorem as follows [4]: Let F be nonnegative and primitive as we had proved. Then, F possessesan eigenvalue r (called the Perrun-Frobenius eigenuafue) such that: 1. T is simple, real, and positive, and r> I?) for any other eigenvalue 7. 2. There exist left and right eigenvectors 5? and 9’ corresponding to the eigenvalue T, which have strictly positive coordinates. 3. There exists a constant matrix D which satisfies =D>O,

where D =9;sj, i, j U, and the eigenvectors are normalized so that 9” = Cfr i 5$%‘~= 1.

Appendix D. Queue length distribution We derive the probability matrix for the queue state in an arbitrary time slot. Let x/(i) be the

1930

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conditional probability that the queue length is i given that the source has a phase j at time slot k, which is given by X/(i) = Pr(X, = ilG, = j}, and let XL(i) be its corresponding matrix x,(i)

=(X:(i),

...3Xli”Ci))‘.

(33)

Let us also define the conditional probability for the cell arrival by y’(i) = Pr(Y, = i\G, = j), and F’(i) = Pr(Y, 2 ilG, = j}. Let the matrix Y(i) = diag(y’(i),y?i), . . . , y”(i))PT, where P = (Pi,j) = Pr{G, = jlGt-, = i], is the state transition matrix for the phase of the source between the (k - 1)th and kth time slot. Let the resulting matrix for the arrival . . . ,jj”(i))PT. process Y”(i) = diag(y’(i),F*(i), Then, the probability matrix x,(i) is given by c- 1 xdi) = Y(i) C x,-,(j) j=O

+ fj

Y(i-j+c)Xk-,(j),OSi
j=c

c-l XkW=f(K)c Xk-t(i) i=O

f i

?(K-i+c)Xkml(i),

i= K,

i=c (34)

which can be solved recursively if the initial condition is given. If we reflect the initial condition in Section 2, we may have

/y,(i)= eT

i = 0,

(35) otherwise, where e is the 1 X h4 matrix with all elements equal to one and 0 is the 1 X M matrix in which all components equal to zero. OT

References [I]

H.S. Bradlow, Performance Measures for Real-Time Continuous Bit-Stream Oriented Services: Application to packet reassembly, Computer Networks and ISDN Systems 20 (1990) 15-26. [2] J.A. Bucklew, Large Deviation Techniques in Decision, Simulation, and Estimation, John Wiley and Sons, Inc., New York, 1990.

and ISDN Systems 29 (I 997) I91 9-1931 [3] C.S. Chang, Stability, Queue Length, and Delay of Detenninistic and Stochastic Queueing Networks, IEEE Tr. Auto. Cont. 39 (5) (1994) 913-931. [4] A. Dembo, 0. Zeitouni, Large Deviation Techniques and Applications, Jones and Bartlett Publishers, Inc.. 1993. [5] R.S. Ellis, Large Deviations for a General Class of Random Vectors, The Ann. of Prob. 12 (1) (1984) I-12. [6] A.I. Elwalid, D. Mitra, Statistical multiplexing with loss priorities in rate based congestion control of high-speed networks, IEEE Tr. Commu. 42 (11) (1994) 2989-3002. [7] D. Ferrari, Client Requirements for Real-Time Communication Services, IEEE Comm. Mag. 2811 (1990) 65-72. [8] J. GZrtner, On Large Deviations from the invariant measure, Theo. Prob. Appl. 12 (I) (1977) 24-39. [9] R. Guerin, H. Ahmadi, M. Naghshineh, Equivalent Capacity and its Application to Bandwidth Allocation in High-Speed Networks, IEEE SAC. 9 (7) (1991) [IO] R.A. Horn, CA. Johnson, Matrix Analysis, Cambridge University Press, 1985. [I I] J.Y. Hui, Switching and Traffic Theory for Integrated Broadband Networks, Kluwer Academic Publishers, 1990. [12] J.Y. Hui, Resource Allocation for Broadband Networks, IEEE SAC. 69 (1988) 1598-1608. [13] G. Kesidis, J. Walrand, C.S. Chang, Effective Bandwidths for Multiclass Markov Fluids and other ATM Sources, IEEE/ACM Tr. Networking 1 (4) (1993) 424-428. [14] J.F.C. Kingman, Inequalities in the Theory of Queues, J. Roy. Stat. Sot. 32 (Series B) (1970) 102-I 10. [15] H. Kobayashi, A.G. Kouheim, Queueing Models for Computer Communications System Analysis, IEEE Tr. Comm. COM-25 (1) (1977) 2-29. [16] J. Kurose, Open Issues and Challenges in Providing Qualityof-Service Guarantees in High-Speed Networks, ACM Computer Communication Review 23 (1) (1993) 6-15. (171 J.-Y. Le Boudec, An efficient solution method for Markov models of ATM links with toss priorities, IEEE Jr. on SAC. 9 (3) (1990 408-417. [I81 Hoon Lee, Yoshiaki Nemoto, Lossy periods in ATM output multiplexer, To be published in IEE? Proceedings-Communications. 1191 S.Q. Li, Study of information loss in packet voice systems, IEEE Tr. on Communications 37 (11) (1989) 1192-1202. 1201 P.A.P. Moran, An Introduction to Probability Theory, Clarendon Press, Oxford, 1968. [21] R. Nagarajan, J. Kurose, D. Towsley, Local allocation of End-to-End Quality-of-Service in High-Speed Networks, Technical Report of Univ. of Massachusetts, TR92-77. [22] V. Ramaswamy, W. Willinger, Efficient Traffic Performance Strategies for Packet Multiplexers, Computer Networks and ISDN Systems 20 (1990) 401-407. [23] J.W. Roberts, J.T. Virtamo, The Superposition of Periodic Cell Arrival Streams in an ATM Multiplexer, IEEE Tr. Comm. COM-39 (2) (1991) 298-303. [24] S.M. Ross, Stochastic Processes John Wiley and Sons, 1983. 1251 S.M. Ross, Bounds on the Delay Distribution in GI/G/I Queues, J. Appl. Prob. I1 (1974) 417-421.

H. Lee. Y. Nemolo / Computer

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[26] 0. Yaron, M. Sidi, Performance and Stability of Communication Networks via Robust Exponential Bounds, IEEE/ACM Tr. Networkin:: 1 (3) (1993) 372-385. Hoon Lee was born in Daegu, Korea, on December 20, 1961. He received the B.E. and M.E. in electronics and communications from KyongPook National University, Daegu, Korea, in 1984 and 1986, respectively. In February 1986, he joined Korea Telecom Research and Development Group, where he has been engaged in the research on the teletraffic theory, network planning, traffic modelling and management of ISDN and B-ISDN networks. He received Ph.D. in electrical and communication engineering from Tohoku University, Sendai. Japan, in 1996. He is now engaged in the research work on the stochastic modelling of a queue, queue input and output control. performance modelling for the ATM networks, QoS guarantee and congestion control in broadband wired and wireless networks. Dr. Lee is a member of IEEE and the Korea Institute of Telematics and Electronics (K.ITE).

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Yashiaki Nemoto was born in Sendai city, Miyagi prefecture, Japan, on December 2, 1945. He received the B-E., M.E., and Dr. Eng. degrees from Tohoku University, Sendai, Japan, in 1968, 1970, and 1973, respectively. From 1973 to 1984 he was a research associate with the Faculty of Engineering, Tohoku University. From 1984 to 199L, he was an Associate Professor with the Research Institute of Electrical Commumcation in the same university. Since 1994, he has been a Professor at Computer Center and at present he is a Professor in the Graduate School of Information Sciences, Tohoku University. He has been engaged in the research work on microwave network, communication systems, computer networking, image processing, hand written character recognition and computer network management. In 1982. he received the Microwave Prize from the IEEE MTT-society. Dr. Nemeto is a member of the IEEE, the Institute of Electronics, Information and Communication Engineers in Japan, and the Information Processing Society of Janan.